Question

prove that x^3-3x+3 is irreducible over Q of rational number

Answer 1

which course are you?

Answer 2

Rational Root Theorem seems the quickest way, no matter the course. The only Rational roots, if there are any, must be 1 or 3.

Answer 3

@tkhunny I would like to know the course in case it is required linear algebra proof. I mean proving based on the basis of P_3

Answer 4

it is abstract algebra

Answer 5

oh yea!! that's what I think. hehehe... It's over my head. I am sorry.

Answer 6

I would be interested as well, but I'd still try the Rational Root Theorem, first.

Answer 7

Okay.

x^3-3x+3

1) 3, a prime number, divides all the coefficients besides the first.
2) 3^2 = 9 does not divide the constant term.

By the Eisenstein Criterion, this is irreducible in Q[x].

Seriously, why do it any hard way?

Answer 8

he gives us hint that prove that it is irreducible over the integers, and use the lama ;let R a unique factorization domain,and let F be the filed of quotients foe R.Then if g(x)is irreducible over R[x] and F[x],then g(x0 is prime in R[x]

Answer 9

Well, okay, but if you use the Eisenstein Criterion, the rest of the class might throw things at you. It will be fun!

Answer 10

3^2 = 9 does not divide the constant term. can u explain ?

Answer 11

why u said 3^2?

Answer 12

It's the criteria. That's all. Ask Eisenstein. :-)